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5.1 Ground state crust

The EoS of the outer crust in the ground state approximation (see Section 3.1) is rather well established, so that the pressure at any given density is determined within a few percent accuracy  [42Jump To The Next Citation Point183Jump To The Next Citation Point357Jump To The Next Citation Point] as can be seen in Figure 27View Image.
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Figure 27: EoS of the ground state of the outer crust for various nuclear models. From Rüster et al. [357Jump To The Next Citation Point]. A zoomed-in segment of the EoS just before the neutron drip can be seen in Figure 28View Image.
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Figure 28: EoS of the ground state of the outer crust just before neutron drip for various nuclear models. From Rüster et al. [357Jump To The Next Citation Point].

Uncertainties arise above the density 10 −3 ρ > 6 × 10 g cm as shown in Figure 28View Image because experimental data are lacking. However, one may hope that with the improvement of experimental techniques, experimental data on very exotic nuclei will become available in the future.

On the contrary, the inner crust nuclei cannot be studied in a laboratory because their properties are influenced by the gas of dripped neutrons, as reviewed in Section 3.2. This means that only theoretical models can be used there and consequently the EoS after neutron drip is much more uncertain than in the outer layers. The neutron gas contributes more and more to the total pressure with increasing density. Therefore, the problem of correct modeling of the EoS of a pure neutron gas at subnuclear densities becomes important. The true EoS of cold catalyzed matter stems from a true nucleon Hamiltonian, expected to describe nucleon interactions at ρ ≲ ρ0, where ρ0 is the nuclear saturation density. To make the solution of the many-body problem feasible, the task is reduced to finding an effective nucleon Hamiltonian, which would enable one to calculate reliably both the properties of laboratory nuclei and the EoS of cold catalyzed matter for 1011g cm −3 ≲ ρ ≲ ρ 0. The task also includes the calculation of the crust-core transition. We will illustrate the general results with two examples of the EoS of the inner crust, calculated in the compressible–liquid-drop model (see Section 3.2.1) using the effective nucleon-nucleon interactions FPS (Friedman–Panharipande–Skyrme [320]) and SLy (Skyrme–Lyon [868887]).

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Figure 29: Comparison of the SLy and FPS EoSs. From [184Jump To The Next Citation Point].

As one can see in Figure 29View Image, significant differences between the SLy and FPS EoS are restricted to the density interval 4 × 1011 – 4 × 1012 g cm–3. They result mainly from the fact that the density at which neutron drip occurs for each is different: ρND (SLy ) ≃ 4 × 1011 g cm −3 (in good agreement with the “empirical EoS” of Haensel & Pichon [183Jump To The Next Citation Point]), while ρND (FPS ) ≃ 6 × 1011 g cm −3. For 12 −3 14 −3 4 × 10 g cm ≲ ρ ≲ 10 g cm the SLy and FPS EoSs are very similar, with the FPS EoS being a little softer at the highest densities considered. The detailed behavior of the two EoSs near the crust-core transition can be seen in Figure 30View Image. The FPS EoS is softer there than the SLy EoS (for pure neutron matter the FPS model is softer at subnuclear densities; see [184Jump To The Next Citation Point]).

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Figure 30: Comparison of the SLy and FPS EoSs near the crust-core transition. Thick solid line: inner crust with spherical nuclei. Dashed line corresponds to “exotic nuclear shapes”. Thin solid line: uniform npe matter. From [184Jump To The Next Citation Point].

In the case of the SLy EoS, the crust-liquid core transition takes place as a very weak first-order phase transition, with a relative density jump on the order of one percent. Notice that, for this model, spherical nuclei persist to the very bottom of the crust [126Jump To The Next Citation Point]. As seen from Figure 30View Image, the crust-core transition is accompanied by a noticeable stiffening of the EoS. For the FPS EoS the situation is different. Namely, the crust-core transition takes place through a sequence of phase transitions with changes of nuclear shapes as discussed in Section 3.3. These phase transitions make the crust-core transition smoother than in the SLy case, with a gradual increase of stiffness (see Figure 35View Image). While the presence of exotic nuclear shapes is expected to have dramatic consequences for the transport, neutrino emission, and elastic properties of neutron star matter, their effect on the EoS is rather small.

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Figure 31: Adiabatic index γ for various EoSs of the ground-state outer crust below neutron drip. The horizontal line corresponds to γ = 4 ∕3. The neutron drip point ρND ≈ 4 × 1011 g cm–3 depends slightly on the EoS model used and, therefore, is not marked. From Rüster et al. [357Jump To The Next Citation Point].
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Figure 32: Adiabatic index γ for the EoS of the ground-state crust. Dotted vertical lines correspond to the neutron drip and crust-core interface points. Calculations performed using the SLy EoS of Douchin & Haensel [125Jump To The Next Citation Point].
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Figure 33: The SLy EoS. Dotted vertical lines correspond to the neutron drip and crust-core transition. From [184Jump To The Next Citation Point].

The overall SLy EoS of the crust, calculated including adjacent segments of the liquid core and the outer crust, is shown in Figure 33View Image. In the outer crust segment, the SLy EoS cannot be visually distinguished from the EoSs of Haensel & Pichon [183Jump To The Next Citation Point] or Rüster et al. [357], which are based on experimental nuclear masses. An important dimensionless parameter, measuring the stiffness of an EoS at a given density, is the adiabatic index,

d log P nb dP γ = d-log-n--= P-dn-- , (80 ) b b
which at subnuclear density can be approximated by γ ≃ (ρ∕P )dP ∕d ρ. The total pressure P is defined by Equation (24View Equation). The adiabatic index γ is shown in Figures 31View Image and 32View Image as a function of mass density ρ. At ρ > ρ ≳ 109 g cm −3 ND, we have γ ≈ 4∕3. This is because in these outer crust layers, pressure is very well approximated by the sum of the contribution of the ultrarelativistic electron gas (Pe) and of the lattice contribution (PL), which both have the same density dependence 4∕3 ∝ ρ (see Section 3.1). One notices in Figure 32View Image a dramatic softening in the density region following the neutron drip point, ρ ≳ ρND. This means, in particular, that no stable stars can exist with central densities around ρ ND because the compressibility of the matter is too low. Then the EoS stiffens gradually, with a significant increase of γ near the crust-core interface. A jump in γ on the core side is connected with the disappearance of nuclei, and a subsequent stiffening (due to the nucleon-nucleon interaction) in the uniform npe liquid. Figures 31View Image and 32View Image show that the EoS of the inner crust is very different from the polytropic form P ∝ ργ0, where γ 0 is a constant. Tabulated and analytical EoSs of the ground state crust are available online [210118].
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